This is a 64x64 virtual array of "cells", each running a 6502 processor with 1k of RAM.
At every time-step, one of the cells gets its memory mapped into 0x0000-0x03FF.
A random adjacent cell gets its memory mapped into 0x0400-0x07FF.
Then the CPU executes a random number of instructions before an interrupt occurs.
There's some other VM/OS tomfoolery where registers are saved/loaded from zero page, but that's about it.

The main 64x64 board shows one character per cell (just a simple hash of the 1kb cell contents).
The bitmaps to the right of the board show the memory maps of the currently selected cells.
(Most viable self-replicators only copy zero page, so only the top quarter of the bitmaps tend to change.)

The entire board is randomly seeded from a "soup" of opcodes. Sometimes, self-replicating programs emerge :)
The board is initialized by default with one such self-replicating seed; you can "discover" more by clicking the Mine button.